On refinement of Jackson type inequalities for approximation of differentiable periodic functions by linear operators (Q1909807)

The aim of this paper is the following Theorem. If \(p\in \{1, \infty\}\), \(n\in \mathbb{N}:= \{1, 2, \dots \}\) then for any odd \(r\in \{1, 3, 5, \dots\}\) and \(f\in L^r_p\) there exists \(d_r>0\) such that \[ |f- F_{n,r} (f) |_p\leq K_r n^{-r} \{2^{-1} d_r \omega_{p,1} (f^{(r)}; \pi n^{-1})+ 4^{-1} (1-d_r) \omega_{p,2} (f^{(r)}; \pi n^{-1})\} \] where \(F_{n,r} (f, .)\) and \(K_r\) are the operator and the constant of Favard, \(L^r_p\) is the set of all \(2\pi\)-priodic functions \(f\) whose \((r-1)\)-th derivatives are locally absolutely continuous on the whole axis with \(f^{(r)}\in L_p\) and \(\omega_{p,k} (f;.)\) is the modulus of smoothness of order \(k\in 1, 2,\dots\) in the space \(L_p\).